The 40-year-old Kurd, who had been living with his family for more than two months at the squalid camp, died of head injuries Thursday, the officials said.
According to the Greek police, the man was injured after falling from a scaffolding outside his tent "onto the side of a passing police car," AFP reported.
The incident sparked riots Monday in the camp, with a group of migrants stonning a police van, accusing the police of having deliberately run the man over.
The Greek police used tear gas to disperse the crowd.
Tensions are running high at the makeshift camp, where over 10,000 people have been stranded since Balkan states closed their borders to them several weeks ago, preventing them continuing their journey to northern Europe.
Late Friday night, scuffles erupted between Afghans and Pakistanis near the camp. Four Pakistanis were treated for minor injuries.
The flow of migrants across the Aegean Sea into Greece has slowed significantly since a deal between the EU and Turkey to send irregular migrants back across the water came into effect last month.
According to official numbers, only 53 migrants have reached the Greek Aegean islands from Turkey in the last 48 hours.
Greek Prime Minister Alexis Tsipras Friday urged NATO to expand its anti-trafficking naval force in the Aegean, despite Turkish objections.